Notion Of Duality Research Articles

Isotropic and coisotropic subvarieties of Grassmannians

We generalize the notion of coisotropic hypersurfaces to subvarieties of Grassmannians having arbitrary codimension. To every projective variety X, Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic hypersurfaces in different Grassmannians. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on conormal spaces, which we use as the starting point for our generalization. We also study the dual notion of isotropic varieties by imposing rank one conditions on tangent spaces instead of conormal spaces.

The Effectiveness of Varied Forms of Education in Attitudes and Behaviors Towards Veganism


 
 
 
 In today’s society, the number of people who are socially conscious is rising as information becomes increasingly available. Veganism is one expression of social consciousness and while a diet that abstains from animal products dates back to the days of Native Americans, a poll conducted by Gallup found that only three percent of the United States population is vegan (McCarthy).
 Many studies previously conducted have synthesized the reasons why people choose to be vegan, but none of them have touched on how to best shift attitudes and behaviors of non-practitioners. According to a Vomad study, 68% of vegans were abstaining from consuming animals due to the ill-treatment of animals by society. This means that a large population of practitioners adopted veganism through becoming educated about the treatment of animals.
 The origins of veganism are rooted in Native Americans’ notions of duality with nature. Modern philosophical theories regarding the treatment of animals which are intended to educate society includes Ecological Animalism, focused on dualism between humans and nature, and Ontological Veganism, focused on the equality of all living beings and the morality of inflicting pain on others. People for the Ethical Treatment of Animals (PETA), a pro-vegan organization, has chosen to educate through polarizing marketing and social media campaigns.
 This poses my research question: How effective is education on the philosophies of veganism in influencing non-vegans’ attitudes and behaviors as compared to education through polarizing marketing and social media campaigns?
 
 
 

On Hollow acts

In this paper, we complete an early result by R. Khosravi and M. Roueentan on the hollow act and obtained new properties and characterizations for this notion continues the account of hollow act for commutative monoids begun on 2019. This notion (hollow acts) represents a dual notion to the uniform acts which were submit by M. Roueentan and M. Sedaghatjoo on 2017. An S-act Ms is referred to as hollow act if every subact Ns of Ms is small where a subact Ns is called small (or superfluous) in Ms if for every subact Hs of MS, NS∪HS=MS implies Hs=Ms. Equally, we reformulate this definition in another words as follows: an S-act Ms is referred to as hollow if whenever N1,N2 are subacts of Ms and N1 ∪N2 = Ms implies either N1=Ms or N2= Ms. Conditions under which subacts are inheriting the property of the Hollow act have been examined. As well as the condition on the quotient subact to be Hollow act was studied. Other properties of the small subact differ from those properties which were early studied by the author are investigated. In the interests of simplicity, the relationship between hollow acts and cyclic act is considered. As a consequence, conditions to coincide those classes are shown. Ultimately, the notion of the Hollow act was used to study when the endomorphism monoid will be local.

Symmetry of the phononic landscape of twisted kagome lattices across the duality boundary

In this work, we investigate the symmetry of the phonon landscape of twisted kagome lattices across their duality boundary. The study is inspired by recent work by Fruchart et al. [Nature, 577, 2020], who specialized the notion of duality to the mechanistic problem of kagome lattices and linked it to the existence of duality transformations between configurations that are symmetrically located across a critical transition point in configuration space. Our first goal is to elucidate how the existence of matching phonon spectra between dual configurations manifest in terms of observable wavefield characteristics. To this end, we explore the possibility to aggregate dual kagome configurations into bi-domain lattices that are geometrically heterogeneous but retain a mechanically homogeneous response. Our second objective is to extend the analysis to structural lattices of beams, which are representative of realistic cellular metamaterials. We show that, in this case, the symmetry of the phonon landscape across the duality boundary is broken, implying that the conditions for dual behavior do not merely depend on the geometry, but also on the dominant mechanisms of the cell, suggesting a dichotomy between geometric and functional duality.

Auslander-Reiten Conjecture in a Dual Vein

We consider a dual notion of the famous Auslander-Reiten Conjecture in case of Noetherian algebras over commutative Noetherian rings. Firstly, in the introduction, we will examine its relevance by showing that in an standard situation, the validity of this dual implies the validity of the Auslander-Reiten Conjecture itself. Moreover, in two important cases these two notions coincide: Artin algebras, and Noetherian algebras over complete local Noetherian rings. In this regard we will prove the following theorem: Let $(R, \mathfrak {m})$ be d-Gorenstein, d ≥ 2, and let Λ be a Noetherian R-algebra which is Gorenstein and (maximal) Cohen-Macaulay as R-module. If M is an Artinian self-orthogonal Gorenstein injective Λ-module such that ${\text {Hom}}_{\Lambda }({\Lambda }_{\mathfrak {p}}, M)$ is an injective ${\Lambda }_{\mathfrak {p}}$ -module for every nonmaximal prime ideal $\mathfrak {p}$ of R, then M is injective. Some applications are discussed afterwards.

Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky.
 The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space.
 The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

Initial fundamentals for the generation of a pre model of phenomenological analysis focused on the concept of violence

The article proposes an investigation for the construction of a pre-analysis model, based on the phenomenological conception from the psychology referred to by Fisas Vicenç, to then address the construct of violence, based on the cultural perspective from Rojas Blanco, stripping himself from the dualistic conception of gender that arises from the Cartesian philosophical conception, aspect that prevents addressing the issue of violence as a great psychosocial phenomenon, leaving it in a purely legal problem, without transcending the psychological as an alternative solution to the current problem. The methodology used is of a hermeneutical nature, focusing the interpretation on the theoretical position, the statistics are analyzed, as the interpretative scenario related to the topic, using the main elements of the pre-model as a form of empirical application of the same, to the social contexts that require it. For this, then, the interpretative analysis of the primary elements of the family violence category is used, in relation to the various manifestations of this, at the social level. Essentially, the pre-model, its elements and the way of articulating them to the phenomenological world are presented as a result, for a more objective view at the reality of violence in the different scenarios. It is important to refer to the need to transform the perspective of violence from the complex, dissolving the notion of duality that stands around this category at the level of national and international institutions. A more humane and in-depth look at this category allows more objective decisions to be made, about the different issues that emerge around the concept of violence.

Hopf-Frobenius Algebras and a Simpler Drinfeld Double

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In addition, we show that this construction is unique up to an invertible scalar. Due to this fact, Hopf-Frobenius algebras provide two canonical notions of duality, and give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. We use this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double, but has a much simpler presentation.

Pure Elements and Dual Notions of Prime Elements in Lattice Modules

This paper deals with the pure elements and the dual notions of prime elements (that is, second elements). For this, it introduces the definitions of second element and coprime element. Then it is shown that the concepts of the second element and coprime element are equivalent. Moreover, this study gives us a characterization of comultiplication modules. Finally, it defines pure elements and obtains the relation among pure, idempotent and multiplication elements.

Uniquely separable extensions

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only if the H-depth is 1 (H-separable extension). Similarly, a split extension has a bimodule projection; this idempotent is full if and only if the ring extension has depth 1 (centrally projective extension). Separable and split extensions have separability idempotents and bimodule projections in 1–1 correspondence via an endomorphism ring theorem in Section 3. If the separable idempotent is unique, then the separable extension is called uniquely separable. A Frobenius extension with invertible E-index is uniquely separable if the centralizer equals the center of the over-ring. It is also shown that a uniquely separable extension of semisimple complex algebras with invertible E-index has depth 1. Earlier group-theoretic results are recovered and related to depth 1. The dual notion, uniquely split extension, only occurs trivially for finite group algebra extensions over complex numbers.

On partial cubes, well-graded families and their duals with some applications in graphs

Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the context of these classes of set systems and then use the obtained results for studying graphs. More specifically, we are concerned with the characterization of the finite set systems which themselves and their dual systems are both well-graded, extremal or maximum. On the way to this goal, and maybe also of independent interest, we study the structure of the well-graded families with the property that the size of the system is not much bigger than the size of its essential domain, that is, the set of elements of the domain which are shattered by the system as single element subsets. As another target of the paper, we use the above results to characterize graphs whose set systems of open or closed neighbourhoods, cliques or independent sets are well-graded, extremal or maximum. We clarify the relation of such graphs to the celebrated half-graphs. Through the paper, we frequently relate our investigations to the VC-dimension of the systems. Also we use one-inclusion graphs associated to set systems as an important technical tool.

Generalized 2-absorbing and strongly generalized 2-absorbing second submodules

Let \(R\) be a commutative ring with identity. A proper submodule \(N\) of an \(R\)-module \(M\) is said to be a 2-absorbing submodule of \(M\) if whenever \(abm \in N\) for some \(a, b \in R\) and \(m \in M\), then \(am \in N\) or \(bm \in N\) or \(ab \in (N :_R M)\). In [3], the authors introduced two dual notion of 2-absorbing submodules (that is, 2-absorbing and strongly 2-absorbing second submodules) of \(M\) and investigated some properties of these classes of modules. In this paper, we will introduce the concepts of generalized 2-absorbing and strongly generalized 2-absorbing second submodules of modules over a commutative ring and obtain some related results.

Mutual Determination, Concrescence and Transition. Whitehead’s Speculative Conception of Temporal Subjectivity Interpreted from a Merleau-Pontyan Standpoint

The interpretive approach adopted in this paper is influenced by Merleau-Ponty’s philosophy and in particular by his understanding of Nature, which in turn takes into consideration Whitehead’s work. Whitehead’s philosophy of organism is seen by its author as the metaphysical generalization of problems found in his investigation of natural knowledge. Whitehead admits that a speculative approach is necessitated by the very questions arising from the mathematical concepts of the material world and the revolutions undergone in logic, mathematics and physics at the turn of the century.Whitehead’s understanding of nature is framed from the beginning in terms of a processual approach. However, this notion of process is not fully worked out in the epistemological works and requires a metaphysical deepening. This is due to the fact that the notion of duration adopted in the epistemological works is not sufficient to convey the notion of process. This lack of adequacy is coupled by Whitehead with the need to interpret process in terms of experience. In turn, this notion of experience is wider than the usual one, for it implies that there is experience from the lowest levels onwards. Matter itself experiences. Seen in this perspective, reality is thus conceived in terms of a whole in constant change, whose parts are in mutual connection. This conception derives from Whitehead’s criticism of Aristotle’s substantialism and from his preference for a relationist ontology. The outcome of this approach is a speculative conception of reality in terms of a twofold notion of process: concrescence and transition, which Whitehead sees as the two faces of the creative advance of nature. This dual notion of process is interpreted in this essay in a merleau-pontyan perspective.

The Logical Theory of Canonical Maps: The Elements & Distinctions Analysis of the Morphisms, Duality, Canonicity, and Universal Constructions in Sets

Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, canonicity, and universal constructions in Sets, the category of sets and functions. The analysis extends directly to other Sets-based concrete categories (groups, rings, vector spaces, etc.). Elements and distinctions are the building blocks of the two dual logics, the Boolean logic of subsets and the logic of partitions. The partial orders (inclusion and refinement) in the lattices for the dual logics define morphisms. The thesis is that the maps that are canonical in Sets are the ones that are defined (given the data of the situation) by these two logical partial orders and by the compositions of those maps.

The Complexity of Radicals and Socles of Modules

This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of Z-modules, where the radical of a Z-module M is defined as the intersection of pM={px:x∈M} with p taken from all primes. It shows that ACA0 is equivalent to the existence of radicals of Z-modules over RCA0. We then study socles of modules over commutative rings with identity. The socle of an R-module M is the largest semisimple submodule of M. We show that the existence of socles of modules over a commutative ring with identity is equivalent to ACA0 over RCA0. Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.

Extended S-supplement submodules

In this paper, we introduced and studied extended S-supplement submodules. A submodule U of a module V is called extended S-supplement submodule in V if there exists a submodule T of V such that V = T + U and U ∩ T is Goldie torsion. Extended S-supplement submodule is a dual notion of extended S-closed submodule. The class of extended S-supplement sequences is a proper class which is generated by nonsingular modules injectively. We studied coinjective objects of this class. Moreover, extended S-supplemented modules are also investigated. We present new characterizations of Z2(RR) -semiperfect rings and SI-rings by extended S-supplement submodules.

Differential superordination for harmonic complex-valued functions

In a previously published paper,in 2015, S. Kanas introduced the differential subordination for harmonic complex-valued functions. In this article,we introduce the dual notion of differential superordination for harmonic complex-valued functions.

Generalized SSP-modules

We say an R-module M has the generalized summand sum property (GSSP), if the sum of any two direct summands is isomorphic to a direct summand of M. This is dual notion to the generalized summand intersection property of modules, and also the class of modules having the GSSP is a proper generalization of the class of modules having the SSP. In this note, the characterization of this property over rings and modules is investigated.

S(0)-Weakly second submodules

We introduce the dual notions of [Formula: see text]-weakly prime submodules, that is, [Formula: see text]-weakly second submodules in a commutative ring with identity. We investigate the properties of [Formula: see text]-weakly second submodules and obtain some related useful characterizations of this dualization. Moreover, we also prove some properties of [Formula: see text]-weakly second submodules related to multiplication and comultiplication modules.

Quantitative program reasoning with graded modal types

In programming, some data acts as a resource (e.g., file handles, channels) subject to usage constraints. This poses a challenge to software correctness as most languages are agnostic to constraints on data. The approach of linear types provides a partial remedy, delineating data into resources to be used but never copied or discarded, and unconstrained values. Bounded Linear Logic provides a more fine-grained approach, quantifying non-linear use via an indexed-family of modalities. Recent work on coeffect types generalises this idea to graded comonads, providing type systems which can capture various program properties. Here, we propose the umbrella notion of graded modal types, encompassing coeffect types and dual notions of type-based effect reasoning via graded monads. In combination with linear and indexed types, we show that graded modal types provide an expressive type theory for quantitative program reasoning, advancing the reach of type systems to capture and verify a broader set of program properties. We demonstrate this approach via a type system embodied in a fully-fledged functional language called Granule, exploring various examples.